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An implicit equation, such as our motivating example
may not be expressed as a function g,
since for x = 0, y=-1 and y=1 both
satisfy our equation.
All hope is not lost, as our equation may be expressed as the
union of two functions:
We may then graph each function separately, and
then combine the two graphs into a single graph.
Consider the following equation:
whose graph follows:
If this graph were to be separated into a collection
of functions, an infinite number of functions would be needed,
since each function may describe at most one point for each
value of x. For any value of x, an infinite number
of values of y satisfy the equation given.
However, for any finite region of the plane,
a finite number of functions suffice.
Some equations, such as
require an infinite number of functions,
even to graph finite regions of the plane,
using the procedure just described.
Next: 1.3 Relations
Up: 1 Motivation
Previous: 1.1 Sampling